Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle CBD = 8x - 25$, and $ m \angle ABC = 2x - 5$, find $m\angle CBD$. $B$ $A$ $D$ $C$
From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {2x - 5} + {8x - 25} = {90}$ Combine like terms: $ 10x - 30 = 90$ Add $30$ to both sides: $ 10x = 120$ Divide both sides by $10$ to find $x$ $ x = 12$ Substitute $12$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 8({12}) - 25$ Simplify: $ {m\angle CBD = 96 - 25}$ So ${m\angle CBD = 71}$.